1. Let U24 = {z € C|224 = 1}. Define $24 = cos () + i sin (5). %3D (a) List all subgroups of U24 and draw its lattice diagram. (b) Let W = ((§24)20) < U24. What is [U24 : W]? (c) Enumerate all left cosets of W in U24.

Q: Which of the following is correct? * O U(14) is isomorphic to S3 O U(14) is isomorphic to U(12)…

A: (a) U(14) is isomorphic to S3. Consider U(14). The elements in U(14) are the elements that are…

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A: please see the next step for solution

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Q: 9. Describe the group of the polynomial (x* – 1) e Q[x] over Q.

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Q: 12. Describe the group of the polynomial (x* – 5x? + 6) € Q[x] o over Q.

A: please see the next step for solution

Q: 1. Consider H= (i) and K= (-1) the subgroups of Q, the quaternion group. 1) Explain why H is normal…

A: Basic theory regarding normal subgroups from algebra will be used

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A: Given that G=ℤ/4ℤ×ℤ/6ℤ.i.e. G=ℤ4×ℤ6H is a subgroup of G generated by 2,2i.e. H=&lt;2,2&gt; oG=4×6=24

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Q: Define on R the operation * by x*y = X+y+k, for all x,y element of R and k is fixed real number. The…

A: We have to check

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A:

Q: 1) Prove that G = {a + bv2 \a,b € Q and a, b # 0} is a subgroup of (R – (0},.)

A: Subgroup

Q: 2. In this exercise, we will construct a new finite group of order 8 using the quater- nions. The…

A: (a)  Recall  that the Quaternion group is  the group which consists   of the  eight quaternions .…

Q: (5,00 5 - Consider the function 0: (R {0}, )→ (R {0}, - ). Then, which of the followings is not a…

A: As per question asked we firstly define the property of homomorphism i.e. if x1, x2 belongs to x…

Q: Compute the indicated values for the indicated homomorphisms. Φ((12)(231)(23)), where Φ: S_3-->Z_2

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Q: 6. Describe the left cosets of SL2(R) in GL₂(R). What is the index of SL₂(R) in GL₂(R)?

A: The left coset for a group G with a subgroup H, tis defined by . Since  can be changed; therefore,…

Q: 1. Let U24 = {z E C|224 = 1}. Define §24 = cos () + i sin (). %3D (a) List all subgroups of U24 and…

A: The given question is related with group theory. Given that ,          U24 = z∈ℂ | z24 = 1 and ξ24 =…

Q: 3. In ROR under componentwise addition, let H = {(r,3r): E R}. (a) Show that H is subgroup of ROR.…

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Q: 3. Let f : (R\{0},-) (R\{0}, -) be group homomorphism defined by f(a) = |a|. Then ker(f) %3D =...

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Q: (a) Make addition and multiplication tables for Z/3Z.

A: “Since you have posted multiple questions, we will provide the solution only to the first question…

Q: Prove that the group G with generators x, y, z and relations z' = z?, x² = x², y* = y? has order 1.

A: In order to solve this question we need to make the set of group G by finding x, y and z.

Q: 4 Let f = (4 5 6) and g = (1984) (2 7 5) (36) be two permutations in S9. (a) Compute fogo f-1, and…

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Q: all possible Jordan Canonical Form of A if the characteristic poly- al is given by (1 – x)*(2 – x)².

A:

Q: 5. Let H= [ (û ¿)I a,b,c EIRI, show that: (H,+) is a sub gromp of (M,+).

A:

Q: 1. Let Sa = {0, sets of the form (a1, b1] × a(Sa) = B, the Borel o-algebra. (ad, bal - o < a; < bị <…

A: We know,  The Borel σ- algebra on R is defined to be generated by open intervals (a, b] for a&lt;b,…

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Q: (4.2) Let X = {1,2, 3, 4} and 7 = {0, {1,2,3} , {2, 3},{1},X} be the topology defined on X. (a) Show…

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Q: 4.6 List all the elements of (Z45,) that are of order 15.

A: (.) Given  group is  ⊕ .(.). The group  ⊕ is an abelian group under addition modulo   and given by ,…

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Q: -. Prove that H = {[17]: nez} is a cyclic subgroup of GL₂(R). IC I 61 10

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Q: 2. Let G = GL(2, R). Prove that the following two subsets of GL(2, R) are subgroups of GL(2, R). (a)…

A: As per the policy we are solving first question only. Please repost the question and specify which…

Q: 12. (Exercise 16.22) Prove the third isomorphism theorem. At each major step in the proof, explain…

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Q: ACR group A = {x+y: −1 < x≤0, 2 < y ≤ 3}. Define group A's infinum (inf A).

A: As  -1 &lt; x And  2 &lt; y Then   -1+2 &lt; x+y I. e. 1 &lt; x+y

Q: 4. If G = (Z8,) then G has an element of order 5. *True *False

A:

Q: (2) In the group G =Z, let H =(4) and K =(8) %3D (i) (ii) List the elements inG /H, GIK and in H/K ,…

A: G=ℤ24H=0,4,8,12,16,20K=0,8,16

Q: Q1/ If (H,*) is collection of subgroups of (G,*) then (U H,*) is subgroup of (G,*)

A:

Q: 1.)Integrate: ∫(x+1)(x+2)^6dx Group of answer choices a.)(2/56) (7x + 6)(x + 2)7 + C b.)(2/56)…

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Q: Let A: l²l² be given by 1 2 {x1,5x2,. 5² A(x1, x2,...) = = Find the spectrum o(A). n 2n + 1 In,

A: As per the question we are given a linear transformation A : l2 → l2 and we have to find its…

Q: f: (A, SA)(B,S) be an order isomorphism and, Sa={xEA: x<naf## and Spa) =y eB= y <gf(@)}. 3) Let %3D…

A: Consider the given information: Let f:A,≤A⇒B,≤B be an order isomorphism. Sa=x∈A: x&lt;Aa≠∅ and…

Q: Which of the following is isomorphism? f:(Z, +) → (Z, +) where f(x) = 2x. this option f: (R, +) →…

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Q: Which of the following is isomorphism? * f:(Z, +) → (R*, .) where f(x) = 2* O this option f: (Z, +)…

A: A mapping is said to be isomorphism if it is one-one, onto and homomorphism. A mapping f: X→Y is…

Q: LET G= R\{-1}. Define * on G by a*b = a + ab+b. Find the solution the solution equation 3* x * 2=23…

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Q: Suppose &lt;let A= 1 and you know all the cofactors in C. How can you find A?

A: Given: Determinant of a matrix A  =  1 and all the co factors are known.   To find: The matrix A.

Q: 4. In the ficld (C,+, ) of complex numbers, define the mapping f: C- C by the rule f(a, b) - (a,…

A: